First, the two limiting cases for length-binning regularization are calculated. These limiting cases are uniform-temperature and non-negative least-squares. This step is much faster than the inversions in step 3, and helps to interpret the inversion results. The best uniform-temperature retrieval is determined using an exhaustive domain search. The least-squares solution is found using MATLAB’s \(lsqnonneg()\)  algorithm [1]}, which outputs the \(H_2O\)  column density PDF at each temperature from 300-1300 Kelvin. If the least-squares solution does not fit the linestrength measurement to at least a 7% reduction in the 2-norm residual relative to the uniform solution, we decide that absorption spectroscopy provides insufficient evidence for temperature nonuniformity, and we select the uniform-temperature solution without performing the Tikhonov regularization algorithm. This 7% guideline is a result of the tendency, as predicted by TSVD in Figure REF a, of truly uniform profiles to produce nonuniform retrievals from absorption spectroscopy measurements with insufficient number of absorption features or SNR.
